Domain scaling and marginality breaking in the random field Ising model

A scaling description is obtained for the $d$--dimensional random field Ising model from domains in a bar geometry. Wall roughening removes the marginality of the $d=2$ case, giving the $T=0$ correlation length $\xi \sim \exp\left(A h^{-\gamma}\right)$ in $d=2$, and for $d=2+\epsilon$ power law behaviour with $\nu = 2/\epsilon \gamma$, $h^\star \sim \epsilon^{1/\gamma}$. Here, $\gamma = 2,4/3$ (lattice, continuum) is one of four rough wall exponents provided by the theory. The analysis is substantiated by three different numerical techniques (transfer matrix, Monte Carlo, ground state algorithm). These provide for strips up to width $L=11$ basic ingredients of the theory, namely free energy, domain size, and roughening data and exponents.